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Inaccessible cardinal
An inaccessible cardinal (or strongly inaccessible cardinal) is a cardinal number that is an uncountable regular strong limit cardinal. The smallest inaccessible cardinal is sometimes called the inaccessible cardinal \(I\). Breaking down the definition, an inaccessible cardinal \(\alpha\) must be: * Uncountable: \(\alpha \geq \omega_1\). * Regular: \(\alpha\) cannot be expressed as the supremum of a set \(S\) of smaller ordinals, where the order type of \(S\) is less than \(\alpha\). * Strong limit: \(\alpha = \beth_\gamma\) for a limit ordinal \(\gamma\), using the following hierarchy of s: ** \(\beth_0 = \aleph_0\) ** \(\beth_{\alpha + 1} = 2^{\beth_\alpha}\) (cardinal exponentiation) ** \(\beth_\alpha = \sup\{\beta < \alpha : \beth_\beta\}\) Examples: *\(\Omega\) is certainly uncountable, and regular, owing to the fact it is the limit of no set with an order smaller than the set of all countable ordinals (which is uncountable, of length \(\Omega\)), but it's corresponding cardinality is \(\aleph_1\), which is not a strong limit cardinal. Therefore \(\Omega\) is not an inaccessible ordinal.. *\(\alpha\mapsto\Omega_{\alpha}\), or the omega fixed point, is uncountable, and a strong limit ordinal, but is not regular, being the limit of \(\{\Omega,\Omega_{\Omega},\Omega_{\Omega_{\Omega}},...\}\), a set with an order of \(\omega\). Therefore \(\alpha\mapsto\Omega_{\alpha}\) is not an inaccessible ordinal. Inaccessible cardinals are far larger than any of the examples given above, as they must quality for all three conditions of being uncountable, regular, and a strong limit. The difference between "strongly" and "weakly" inaccessible cardinals: If we replace "strong limit cardinal" with "limit cardinal" (replacing "beth numbers" with " s"), we get weakly inaccessible cardinals. The distinction between strongly and weakly inaccessible cardinals only matters if we don't assume (GCH). Under GCH, all limit cardinals are strong limit cardinals. Properties *GCH aside, if ZFC is consistent, neither weakly nor strongly inaccessible cardinals can be proven to exist within it. A stronger theory, , can prove their existence. ZFC + "there exists a weakly inaccessible cardinal" is believed to be consistent. *Often, the first inaccessible cardinal (if it exists), denoted \(I\), is considered the threshold for large cardinals. That is, all cardinals less than \(I\) are small, and all cardinals at least \(I\) are large. *\(I_{\omega}\) is not the supremum of \(\{I,I_2,I_3,...\}\), where \(I_n\) is the nth inaccessible cardinal. This follows from the fact that \(I_{\alpha}\) has to be regular, and if \(I_{\omega}=\{I,I_2,I_3,...\}\) it would have order \(\omega\), which is less than \(I_{\omega}\). The only way to construct a list of ordinals whose supremum is \(I_{\omega}\) is to construct a list with a length of \(I_{\omega}\). It must be defined in terms of itself. Collapsing functions using inaccessible cardinals The inaccessible cardinals are most relevant to googology through ordinal collapsing functions. The function \(\alpha \mapsto \psi_I(\alpha)\) enumerates the fixed points of \(\beta \mapsto \Omega_\beta\), so \(\psi_I(0)\) is the omega fixed point, and \(\psi_I(1)\) is the second fixed point of \(\beta \mapsto \Omega_\beta\), and so on. In turn, these extremely large uncountable ordinals can be turned into large countable ordinals by enclosing them in lower order psi expressions (e.g. \(\psi_0(\psi_I(0))\) and \(\psi_0(\psi_I(1))\) respectively), which can then be placed in an ordinal to natural hierarchy such as the FGH (e.g. \(f_{\psi_0(\psi_I(0))}(n)\) and \(f_{\psi_0(\psi_I(1))}(n)\)). See also *Inaccessible cardinal on Cantor's Attic normal function ordinal notation ordinal numbers fundamental sequence |Section 2='Theories:' Presburger arithmetic Peano arithmetic KP second-order arithmetic ZFC |Section 3='Countable ordinals:' \(\omega\) \(\varepsilon_0\) \(\zeta_0\) \(\Gamma_0\) \(\vartheta(\Omega^3)\) \(\vartheta(\Omega^\omega)\) \(\vartheta(\Omega^\Omega)\) \(\vartheta(\varepsilon_{\Omega + 1})\) \(\psi(\Omega_\omega)\) \(\psi(\varepsilon_{\Omega_\omega + 1})\) \(\psi(\psi_I(0))\) \(\omega_1^\mathfrak{Ch}\) \(\omega_1^{CK}\) \(\lambda,\zeta,\Sigma,\gamma\) List of countable ordinals |Section 4='Ordinal hierarchies:' Fast-growing hierarchy Slow-growing hierarchy Hardy hierarchy Middle-growing hierarchy N-growing hierarchy |Section 5='Uncountable cardinals:' \(\omega_1\) omega fixed point inaccessible cardinal \(I\) Mahlo cardinal \(M\) weakly compact cardinal \(K\) indescribable cardinal rank-into-rank cardinal more... }} Japanese Wikia Article:到達不可能基数 Category:Transfinite ordinals